Softcover ISBN:  9780821833995 
Product Code:  CLN/10 
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Electronic ISBN:  9781470417604 
Product Code:  CLN/10.E 
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Book DetailsCourant Lecture NotesVolume: 10; 2003; 323 ppMSC: Primary 35;
The nonlinear Schrödinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or the Kortewegde Vries equation. From the mathematical point of view, Schrödinger's equation is a delicate problem, possessing a mixture of the properties of parabolic and elliptic equations. Useful tools in studying the nonlinear Schrödinger equation are energy and Strichartz's estimates.
This book presents various mathematical aspects of the nonlinear Schrödinger equation. It studies both problems of local nature (local existence of solutions, uniqueness, regularity, smoothing effect) and problems of global nature (finitetime blowup, global existence, asymptotic behavior of solutions). In principle, the methods presented apply to a large class of dispersive semilinear equations. The first chapter recalls basic notions of functional analysis (Fourier transform, Sobolev spaces, etc.). Otherwise, the book is mostly selfcontained.
It is suitable for graduate students and research mathematicians interested in nonlinear partial differential equations and applications to mathematical physics.ReadershipGraduate students and research mathematicians interested in nonlinear partial differential equations and applications to mathematical physics.

Table of Contents

Chapters

Chapter 1. Preliminaries

Chapter 2. The linear Schrödinger equation

Chapter 3. The Cauchy problem in a general domain

Chapter 4. The local Cauchy problem

Chapter 5. Regularity and the smoothing effect

Chapter 6. Global existence and finitetime blowup

Chapter 7. Asymptotic behavior in the repulsive case

Chapter 8. Stability of bound states in the attractive case

Chapter 9. Further results


Additional Material

Reviews

The book, written by one of the leading expert on the subject, is also an upto date source of references for recent results and open problems, well represented in its extensive bibliography. It can certainly be used as a guideline for a course to an audience with a sufficient background on functional analysis and partial differential equations.
Zentralblatt MATH 
In summary, the author gives a well balanced treatment of the many types of mathematical results... This book would be an excellent place to start for readers interested in an introduction to these topics. There is an extensive bibliography which nicely complements the author's discussions.
Woodford W. Zachary for Mathematical Reviews


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The nonlinear Schrödinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or the Kortewegde Vries equation. From the mathematical point of view, Schrödinger's equation is a delicate problem, possessing a mixture of the properties of parabolic and elliptic equations. Useful tools in studying the nonlinear Schrödinger equation are energy and Strichartz's estimates.
This book presents various mathematical aspects of the nonlinear Schrödinger equation. It studies both problems of local nature (local existence of solutions, uniqueness, regularity, smoothing effect) and problems of global nature (finitetime blowup, global existence, asymptotic behavior of solutions). In principle, the methods presented apply to a large class of dispersive semilinear equations. The first chapter recalls basic notions of functional analysis (Fourier transform, Sobolev spaces, etc.). Otherwise, the book is mostly selfcontained.
It is suitable for graduate students and research mathematicians interested in nonlinear partial differential equations and applications to mathematical physics.
Graduate students and research mathematicians interested in nonlinear partial differential equations and applications to mathematical physics.

Chapters

Chapter 1. Preliminaries

Chapter 2. The linear Schrödinger equation

Chapter 3. The Cauchy problem in a general domain

Chapter 4. The local Cauchy problem

Chapter 5. Regularity and the smoothing effect

Chapter 6. Global existence and finitetime blowup

Chapter 7. Asymptotic behavior in the repulsive case

Chapter 8. Stability of bound states in the attractive case

Chapter 9. Further results

The book, written by one of the leading expert on the subject, is also an upto date source of references for recent results and open problems, well represented in its extensive bibliography. It can certainly be used as a guideline for a course to an audience with a sufficient background on functional analysis and partial differential equations.
Zentralblatt MATH 
In summary, the author gives a well balanced treatment of the many types of mathematical results... This book would be an excellent place to start for readers interested in an introduction to these topics. There is an extensive bibliography which nicely complements the author's discussions.
Woodford W. Zachary for Mathematical Reviews